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Hand-out%205 - (see Exercise F1 on page 29 in the notes An...

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The linear additive model Models have been developed in all sciences, including the statistical sciences. A model is defined by one or several equations (with a left-hand side and a right-hand side). Statistical models are useful in making formal calculations (see Assignment 2!), deriving general properties (see Section 1.6!) and/or describing relationships. If X 1 , …, X n denotes a random sample of size n for the random variable X, the linear additive model X i = µ + ε i (i = 1, …, n) is a reduced version of the simple linear regression model (Section 5.7) Y i = a + bx i + ε i (i = 1, …, n), in which Y i = X i , a = µ is the population mean of the random variable X, and b = 0. One of the ‘discoveries’ announced in Hand-out 4 is that the sample mean is a good approximation (later, we shall say estimator ) of the population mean, as the sample mean tends to be closer to the population mean when the sample size n increases
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Unformatted text preview: (see Exercise F1 on page 29 in the notes). An introduction to probability elements Preamble Contrary to physical laws, statistical laws are not deterministic in that you don’t know the outcome of a probabilistic experiment beforehand. Events are possible outcomes of a probabilistic experiment. The number associated with their chance of being observed is their probability . If A and B denote two events of the same probabilistic experiment, then 0 ≤ P(A), P(B) ≤ 1 Key note One does not work with events (which are like sets) as one works with probabilities (which are numbers in the interval [0, 1]). A different type of operators is required for events: union A U B, intersection A ∩ B, and complements A c , B c . Operators for probabilities are the classical arithmetic operators: . addition (+), difference (-), product ( x ), and division by a non-zero quantity (/)....
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